Homework Answers
In this question, the projection of vector x on l is
x*cos
and the mirror image will also have the projection on l as
x*cos
therefore, overall projections of vector x on l are
2*(x*cos )
This is the projections as the x projections along other axes apart from l will cancel each other.
4. Let S be a plane in R3 passing through the origin, so that S is...
4. Let S be a plane in R3 passing through the origin, so that S is a two- dimensional subspace of R3. Say that a linear transformation T: R3 R3 is a reflection about Sif T(U) = v for any vector v in S and T(n) = -n whenever n is perpendicular to S. Let T be the linear transformation given by T(x) = Ar, 1 1 А -2 2 2 21 -2 2 3 T is a reflection about...
Let L be a straight line through the origin and denote by Show that: the angle...
Let L be a straight line through the origin and denote by Show that: the angle from positive r-axis to L. (a) The matrix representing projection to the line L is given by P = ſcos cos O sin sin cos 6] sin0 (b) The matrix representing reflection about the line L is cos 20 sin 20 sin 201 - cos 20
Let L in R 3 be the line through the origin spanned by the vector v...
Let L in R 3 be the line through the origin
spanned by the vector v = 1 1 3 . Find the linear equations
that define L, i.e., find a system of linear equations whose
solutions are the points in L.
lil (6) Let L in R3 be the line through the origin spanned by the vector v= 1. Find the linear equations that define L, i.e., find a system of linear equations whose solutions are...
Let L in R 3 be the line through the origin spanned by the vector v...
Let L in R 3 be the line through the origin spanned by the
vector v = 1 1 3 . Find the linear equations that define L,
i.e., find a system of linear equations whose solutions are the
points in L. (7) Give an example of a linear transformation from T
: R 2 → R 3 with the following two properties: (a) T is not
one-to-one, and (b) range(T) = ...
Let u= -3 2 4 ; and let L denote the line thru the origin of R3 in the direction of u. The proj...
Let u= -3 2 4 ; and let L denote the line thru the origin of R3 in the direction of u. The projection of R3 onto L — denoted PL : R3 −→ R3 — is definded to be equal to the projection pu onto the vector u. You may assume that PL is a linear transformation. Find the standard matrix [PL] for PL.
Let l denote the line through the origin with direction vector (1,1,1). Let r(t) = (t...
Let l denote the line through the origin with direction vector (1,1,1). Let r(t) = (t +1,4, 2t) be a parametrized curve. Compute the point r(to) on the curve which is closest to l, and state the distance from r(to) to l.
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
4. Let l be the line through the origin which has a directed) angle of inclination...
4. Let l be the line through the origin which has a directed) angle of inclination 8 from positive x-axis. Let l' be the line through h + Oi with (directed) angle of inclination from positive c-axis. (a) Use re = ROTOR_, to derive the complex equation for the reflection re(2). (b) Use part (a) and re = T re(2) reo T-to derive the complex equation of the reflection
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1&#...
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1' parallel to through P in the usual way: drop a perpendicular PQ to / and construct /" perpendicular to PQ. Let I" be any line through P distinct from l'. To see that /" intersects I, pick a point A on PQ between...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle betwe...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
4. Let S be a plane in R3 passing through the origin, so that S is a two- dimensional subspace of R3. Say that a linear transformation T: R3 R3 is a reflection about Sif T(U) = v for any vector v in S and T(n) = -n whenever n is perpendicular to S. Let T be the linear transformation given by T(x) = Ar, 1 1 А -2 2 2 21 -2 2 3 T is a reflection about...
Let L be a straight line through the origin and denote by Show that: the angle from positive r-axis to L. (a) The matrix representing projection to the line L is given by P = ſcos cos O sin sin cos 6] sin0 (b) The matrix representing reflection about the line L is cos 20 sin 20 sin 201 - cos 20
Let L in R 3 be the line through the origin
spanned by the vector v = 1 1 3 . Find the linear equations
that define L, i.e., find a system of linear equations whose
solutions are the points in L.
lil (6) Let L in R3 be the line through the origin spanned by the vector v= 1. Find the linear equations that define L, i.e., find a system of linear equations whose solutions are...
Let L in R 3 be the line through the origin spanned by the
vector v = 1 1 3 . Find the linear equations that define L,
i.e., find a system of linear equations whose solutions are the
points in L. (7) Give an example of a linear transformation from T
: R 2 → R 3 with the following two properties: (a) T is not
one-to-one, and (b) range(T) = ...
Let l denote the line through the origin with direction vector (1,1,1). Let r(t) = (t +1,4, 2t) be a parametrized curve. Compute the point r(to) on the curve which is closest to l, and state the distance from r(to) to l.
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
4. Let l be the line through the origin which has a directed) angle of inclination 8 from positive x-axis. Let l' be the line through h + Oi with (directed) angle of inclination from positive c-axis. (a) Use re = ROTOR_, to derive the complex equation for the reflection re(2). (b) Use part (a) and re = T re(2) reo T-to derive the complex equation of the reflection
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1' parallel to through P in the usual way: drop a perpendicular PQ to / and construct /" perpendicular to PQ. Let I" be any line through P distinct from l'. To see that /" intersects I, pick a point A on PQ between...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
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